#7737469

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30);(1,2)(3,8)(4,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(2,7)(3,4)(9,11,12,15)(10,13,14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,3,6,4)(2,8,7,5);(1,4,6,3)(2,5,7,8)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,5)(2,3)(4,7)(6,8)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644665);